Groups of order 72: Difference between revisions
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{{groups of order|72}} | {{groups of order|72}} | ||
{{specific information about this order|72}} | |||
==Statistics at a glance== | |||
= | The number 72 has prime factorization <math>72 = 2^3 \cdot 3^2</math>. {{only two prime factors hence solvable}} | ||
{| class="sortable" border="1" | {| class="sortable" border="1" | ||
! Quantity !! Value !! Explanation | ! Quantity !! Value !! Explanation | ||
|- | |- | ||
| Number of groups | | Number of groups up to isomorphism || [[count::50]] || | ||
|- | |- | ||
| Number of abelian | | Number of [[abelian group]]s || [[abelian count::6]] || (number of groups of order <math>2^3</math>) times (number of groups of order <math>3^2</math>) = ([[number of unordered integer partitions]] of 3) times ([[number of unordered integer partitions]] of 2) = <math>3 \times 2 = 6</math>. {{abelian count explanation}} | ||
|- | |- | ||
| Number of nilpotent | | Number of [[nilpotent group]]s || [[nilpotent count::10]] || (number of [[groups of order 8]]) times (number of [[groups of order 9]]) = <math>5 \times 2 = 10</math>. {{nilpotent count explanation}} | ||
|- | |- | ||
| Number of | | Number of [[solvable group]]s || [[solvable count::50]] || {{only two prime factors hence solvable}} | ||
|- | |- | ||
| Number of simple groups || 0 || Follows from the fact that all groups of the order are solvable. | | Number of simple groups || 0 || Follows from the fact that all groups of the order are solvable. | ||
|} | |} | ||
==Interesting groups== | |||
* [[Mathieu group:M9]] | |||
* [[General affine group:GA(1,9)]] | |||
==GAP implementation== | ==GAP implementation== | ||
Latest revision as of 13:00, 15 November 2023
This article gives information about, and links to more details on, groups of order 72
See pages on algebraic structures of order 72 | See pages on groups of a particular order
This article gives basic information comparing and contrasting groups of order 72. See also more detailed information on specific subtopics through the links:
| Information type | Page summarizing information for groups of order 72 |
|---|---|
| element structure (element orders, conjugacy classes, etc.) | element structure of groups of order 72 |
| subgroup structure | subgroup structure of groups of order 72 |
| linear representation theory | linear representation theory of groups of order 72 projective representation theory of groups of order 72 modular representation theory of groups of order 72 |
| endomorphism structure, automorphism structure | endomorphism structure of groups of order 72 |
| group cohomology | group cohomology of groups of order 72 |
Statistics at a glance
The number 72 has prime factorization . There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.
| Quantity | Value | Explanation |
|---|---|---|
| Number of groups up to isomorphism | 50 | |
| Number of abelian groups | 6 | (number of groups of order ) times (number of groups of order ) = (number of unordered integer partitions of 3) times (number of unordered integer partitions of 2) = . See classification of finite abelian groups and structure theorem for finitely generated abelian groups. |
| Number of nilpotent groups | 10 | (number of groups of order 8) times (number of groups of order 9) = . See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group. |
| Number of solvable groups | 50 | There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order. |
| Number of simple groups | 0 | Follows from the fact that all groups of the order are solvable. |
Interesting groups
GAP implementation
The order 72 is part of GAP's SmallGroup library. Hence, any group of order 72 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.
Further, the collection of all groups of order 72 can be accessed as a list using GAP's AllSmallGroups function.
Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:
gap> SmallGroupsInformation(72);
There are 50 groups of order 72.
They are sorted by their Frattini factors.
1 has Frattini factor [ 6, 1 ].
2 has Frattini factor [ 6, 2 ].
3 has Frattini factor [ 12, 3 ].
4 - 8 have Frattini factor [ 12, 4 ].
9 - 11 have Frattini factor [ 12, 5 ].
12 has Frattini factor [ 18, 3 ].
13 has Frattini factor [ 18, 4 ].
14 has Frattini factor [ 18, 5 ].
15 has Frattini factor [ 24, 12 ].
16 has Frattini factor [ 24, 13 ].
17 has Frattini factor [ 24, 14 ].
18 has Frattini factor [ 24, 15 ].
19 has Frattini factor [ 36, 9 ].
20 - 24 have Frattini factor [ 36, 10 ].
25 has Frattini factor [ 36, 11 ].
26 - 30 have Frattini factor [ 36, 12 ].
31 - 35 have Frattini factor [ 36, 13 ].
36 - 38 have Frattini factor [ 36, 14 ].
39 - 50 have trivial Frattini subgroup.
For the selection functions the values of the following attributes
are precomputed and stored:
IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup,
LGLength, FrattinifactorSize and FrattinifactorId.
This size belongs to layer 2 of the SmallGroups library.
IdSmallGroup is available for this size.